The Restricted Six-Body Problem with Stable Equilibrium Points and a Rhomboidal Configuration

Abstract

We explore the central configuration of the rhomboidal restricted six-body problem in Newtonian gravity, which has four primaries $m_i$ (where $i=1,\dots 4$ ) at the vertices of the rhombus $\left(a,0\right)$ , $\left(-a,0\right)$ , $\left(0,b\right)$ , and $\left(0,-b\right)$ , respectively, and a fifth mass $m_0$ is at the point of intersection of the diagonals of the rhombus, which is placed at the center of the coordinate system (i.e., at the origin $\left(\mathrm{0,0}\right)$ ). The primaries at the rhombus’s opposite vertices are assumed to be equal, that is, $m_1=m_2=m$ and $m_3=m_4=\tilde{m}$ . After writing equations of motion, we express $m_0,m$ , and $\tilde{m}$ in terms of mass parameters $a$ and $b$ . Finally, we find the bounds on $a$ and $b$ for positive masses. In the second part of this article, we investigate the motion and different features of a test particle (sixth body $m_5$ ) with infinitesimal mass that moves under the gravitational effect of the five primaries in the rhomboidal configuration. All four cases have 16, 12, 20, and 12 equilibrium points, with case-I, case-II, and case-III having stable equilibrium points. A significant shift in the position and the number of equilibrium points was found in four cases with the variations of mass parameters $a$ and $b$ . The regions for the possible motion of test particles have been discovered. It has also been observed that as the Jacobian constant $C$ increases, the permissible region of motion expands. We also have numerically verified the linear stability analysis for different cases, which shows the presence of stable equilibrium points.